We develop a new interior point method for solving linear programs. Our algorithm is universal in the sense that it matches the number of iterations of any interior point method that uses a self-concordant barrier function up to a factor $O(n^{1.5}\log n)$ for an $n$-variable linear program in standard form. The running time bounds of interior point methods depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. Our algorithm also admits a combinatorial upper bound, terminating with an exact solution in $O(2^{n} n^{1.5}\log(n))$ iterations. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations.

Following the breakthrough work of Tardos (Oper. Res. '86) in the bit-complexity model, Vavasis and Ye (Math. Prog. '96) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) $\mathrm{max} \\; c^\top x : Ax = b, x \geq 0, A \in \mathbb{R}^{m \times n}$, Vavasis and Ye developed a primal-dual interior point method using a _layered least squares_ (LLS) step, and showed that $O(n^{3.5} \log (\bar\chi_A+n))$ iterations suffice to solve (LP) exactly, where $\bar{\chi}_A$ is a condition measure controlling the size of solutions to linear systems related to $A$. Monteiro and Tsuchiya (SIAM J. Optim. '03), noting that the central path is invariant under rescalings of the columns of $A$ and $c$, asked whether there exists an LP algorithm depending instead on the measure $\bar{\chi}^{\ast}_A$, defined as the minimum $\bar{\chi}_{AD}$ value achievable by a column rescaling $AD$ of $A$, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an $O(m^2 n^2 + n^3)$ time algorithm which works on the linear matroid of $A$ to compute a nearly optimal diagonal rescaling $D$ satisfying $\bar{\chi}_{AD} \le n(\bar{\chi}^\ast)^3$. This algorithm also allows us to approximate the value of $\bar{\chi}_A$ up to a factor $n(\bar{\chi}^\ast)^2$. This result is in (surprising) contrast to that of Tunçel (Math. Prog. '99), who showed NP-hardness for approximating $\bar{\chi}_A$ to within $2^{\mathrm{poly}(\mathrm{rank}(A))}$. The key insight for our algorithm is to work with ratios $g_i/g_j$ of circuits of $A$ — i.e., minimal linear dependencies $Ag=0$ — which allow us to approximate the value of $\bar{\chi}_A^\ast$ by a maximum geometric mean cycle computation in what we call the circuit ratio digraph of $A$. While this resolves Monteiro and Tsuchiya's question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved $O(n^{2.5} \log(n) \log (\bar{\chi}^\ast_A+n))$ iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor $n/\log n$ improvement on the iteration complexity bound of the original Vavasis-Ye algorithm.

Revisiting Tardos's Framework for Linear Programming: Faster Exact Solutions using Approximate Solvers